Advertisements
Advertisements
प्रश्न
The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, −3). The origin is the midpoint of the base. Find the coordinates of the points A and B. Also, find the coordinates of another point D such that ABCD is a rhombus.
Advertisements
उत्तर
Given:
ΔABC is equilateral
Base BC lies on the y–axis
C = (0, –3)
Origin (0, 0) is midpoint of BC
B = (0, b), C = (0, −3)
`0 = (b+(-3))/2`
b − 3 = 0
b = 3
B = (0, 3)
C = (0, –3)
In an equilateral triangle,
height = `sqrt3/2 xx "side"`
Height = `sqrt3/2 xx 6`
`= 3sqrt3`
Base lies on y-axis, so the altitude is horizontal, meaning A lies on the x-axis but at distance `3sqrt3` from O.
`A = (+-3sqrt3, 0)`
A1 = `(3sqrt3, 0)`
A2 = (`-3sqrt3, 0`)
A rhombus has all sides equal, and ABCD is cyclic order.
A = (`3sqrt3`, 0), B = (0, 3), C = (0, −3)
To form a rhombus ABCD, the 4th point D is:
D = A + C − B
D = (`3sqrt3`, 0) + (0, −3) − (0, 3)
D = (`3sqrt3`, −6)
Base points:
B = (0, 3), C = (0, −3)
Vertices of equilateral triangle:
A = (`3sqrt3`, 0) or A = (`−3sqrt3`, 0)
Point D such that ABCD is a rhombus:
If A = (`3sqrt3`, 0):
D = (`3sqrt3`, −6)
If A = (`−3sqrt3`, 0):
D = (`-3sqrt3`, −6)
संबंधित प्रश्न
(Street Plan): A city has two main roads which cross each other at the centre of the city. These two roads are along the North-South direction and East-West direction.
All the other streets of the city run parallel to these roads and are 200 m apart. There are 5 streets in each direction. Using 1cm = 200 m, draw a model of the city on your notebook. Represent the roads/streets by single lines.
There are many cross- streets in your model. A particular cross-street is made by two streets, one running in the North - South direction and another in the East - West direction. Each cross street is referred to in the following manner : If the 2nd street running in the North - South direction and 5th in the East - West direction meet at some crossing, then we will call this cross-street (2, 5). Using this convention, find:
- how many cross - streets can be referred to as (4, 3).
- how many cross - streets can be referred to as (3, 4).
If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k − 1, 5k) are collinear, then find the value of k
Find the co-ordinates of the point equidistant from three given points A(5,3), B(5, -5) and C(1,- 5).
Show that the points A(2,1), B(5,2), C(6,4) and D(3,3) are the angular points of a parallelogram. Is this figure a rectangle?
Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the ratio 7 : 2
`"Find the ratio in which the poin "p (3/4 , 5/12) " divides the line segment joining the points "A (1/2,3/2) and B (2,-5).`
Find the coordinates of circumcentre and radius of circumcircle of ∆ABC if A(7, 1), B(3, 5) and C(2, 0) are given.
Mark the correct alternative in each of the following:
The point of intersect of the coordinate axes is
Points P, Q, R and S divides the line segment joining A(1, 2) and B(6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R.
If the point P (m, 3) lies on the line segment joining the points \[A\left( - \frac{2}{5}, 6 \right)\] and B (2, 8), find the value of m.
If the distance between points (x, 0) and (0, 3) is 5, what are the values of x?
Write the coordinates of the point dividing line segment joining points (2, 3) and (3, 4) internally in the ratio 1 : 5.
Two vertices of a triangle have coordinates (−8, 7) and (9, 4) . If the centroid of the triangle is at the origin, what are the coordinates of the third vertex?
Find the distance between the points \[\left( - \frac{8}{5}, 2 \right)\] and \[\left( \frac{2}{5}, 2 \right)\] .
Find the area of triangle with vertices ( a, b+c) , (b, c+a) and (c, a+b).
Write the X-coordinate and Y-coordinate of point P(– 5, 4)
The points whose abscissa and ordinate have different signs will lie in ______.
Points (1, –1) and (–1, 1) lie in the same quadrant.
